cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A331607 E.g.f.: exp(1 / (1 - sin(x)) - 1).

Original entry on oeis.org

1, 1, 3, 12, 61, 372, 2639, 21280, 191833, 1908688, 20750331, 244478784, 3100597333, 42088689216, 608543191559, 9332562964480, 151252803045937, 2582250195499264, 46306562212010355, 870011934425816064, 17086276243125287917
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2020

Keywords

Crossrefs

Cf. A000111, A000772, A002017, A331608, A331610, A331611.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[1/(1 - Sin[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A000111[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n + 1) (2^(n + 1) - 1) BernoulliB[n + 1])/(n + 1)]]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A000111[k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A000111(k+1) * a(n-k).
a(n) ~ 2^(n + 2/3) * exp(8/(3*Pi^2) - 5/6 + 2^(5/3) * n^(1/3) / Pi^(4/3) + 3 * 2^(1/3) * n^(2/3) / Pi^(2/3) - n) * n^(n - 1/6) / (sqrt(3) * Pi^(n + 1/3)). - Vaclav Kotesovec, Jan 26 2020

A331617 E.g.f.: exp(1 / (1 - arctan(x)) - 1).

Original entry on oeis.org

1, 1, 3, 11, 49, 265, 1683, 12035, 95169, 832337, 7998467, 83033403, 922112305, 10978263257, 139956480467, 1889161216179, 26798589518593, 401123509624737, 6346168059440515, 105040097140558699, 1805102151607613361, 32421358229074354601
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2020

Keywords

Comments

a(53) is negative. - Vaclav Kotesovec, Jan 26 2020

Links

Crossrefs

Cf. A002019, A110708, A191700, A331610, A331615, A331616, A331618.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[1/(1 - ArcTan[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A191700[0] = 1; A191700[n_] := A191700[n] = Sum[Binomial[n, k] If[OddQ[k], (-1)^Boole[IntegerQ[(k + 1)/4]] (k - 1)!, 0] A191700[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A191700[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
  • PARI
    seq(n)={Vec(serlaplace(exp(1/(1 - atan(x + O(x*x^n))) - 1)))} \\ Andrew Howroyd, Jan 22 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A191700(k) * a(n-k).

A331618 E.g.f.: exp(1 / (1 - arctanh(x)) - 1).

Original entry on oeis.org

1, 1, 3, 15, 97, 785, 7523, 83615, 1053281, 14838177, 230832867, 3929944623, 72633052545, 1447981700529, 30960823851267, 706676217730239, 17145815895371073, 440594781536265537, 11952178787661839427, 341291300477569866831, 10231558345117929439521
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2020

Keywords

Crossrefs

Cf. A000246, A080833, A202139, A296676, A331610, A331615, A331616, A331617.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[1/(1 - ArcTanh[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A296676[0] = 1; A296676[n_] := A296676[n] = Sum[Binomial[n, k] If[OddQ[k], (k - 1)!, 0] A296676[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A296676[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
  • PARI
    seq(n)={Vec(serlaplace(exp(1/(1 - atanh(x + O(x*x^n))) - 1)))} \\ Andrew Howroyd, Jan 22 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A296676(k) * a(n-k).
a(n) ~ (exp(2) + 1)^(n - 1/4) * n^(n - 1/4) / ((exp(2) - 1)^(n + 1/4) * exp(n - 4*exp(1)*sqrt(n/(exp(4) - 1)) - 2/(exp(4) - 1) - 1/2)). - Vaclav Kotesovec, Jan 26 2020
Showing 1-3 of 3 results.

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